[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/lander-parkin-and-selfridge-conjecture\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/lander-parkin-and-selfridge-conjecture\/","headline":"Lander, Parkin, and Selfridge conjecture","name":"Lander, Parkin, and Selfridge conjecture","description":"From Wikipedia, the free encyclopedia The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums","datePublished":"2020-02-09","dateModified":"2020-02-09","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fac31bc72cea732d84fe70c844e692021488cbe3","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fac31bc72cea732d84fe70c844e692021488cbe3","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/lander-parkin-and-selfridge-conjecture\/","wordCount":2340,"articleBody":"From Wikipedia, the free encyclopediaThe Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers.  The equations are generalisations of those considered in Fermat’s Last Theorem. The conjecture is that if the sum of some k-th powers equals the sum of some other k-th powers, then the total number of terms in both sums combined must be at least k.Table of ContentsBackground[edit]Conjecture[edit]Current status[edit]See also[edit]References[edit]External links[edit]Background[edit]Diophantine equations, such as the integer version of the equation a2 + b2 = c2 that appears in the Pythagorean theorem, have been studied for their integer solution properties for centuries. Fermat’s Last Theorem states that for powers greater than 2, the equation ak + bk = ck has no solutions in non-zero integers  a, b, c. Extending the number of terms on either or both sides, and allowing for higher powers than 2, led to Leonhard Euler to propose in 1769 that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.In symbols, if\u2211i=1naik=bk{displaystyle sum _{i=1}^{n}a_{i}^{k}=b^{k}}where n > 1 and a1,a2,\u2026,an,b{displaystyle a_{1},a_{2},dots ,a_{n},b} are positive integers, then his conjecture was that n \u2265 k.In 1966, a counterexample to Euler’s sum of powers conjecture was found by Leon J. Lander and Thomas R. Parkin for k = 5:[1]275 + 845 + 1105 + 1335 = 1445.In subsequent years, further counterexamples were found, including for k = 4. The latter disproved the more specific Euler quartic conjecture, namely that a4 + b4 + c4 = d4 has no positive integer solutions.  In fact, the smallest solution, found in 1988, is4145604 + 2175194 + 958004 = 4224814.Conjecture[edit]In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured[2] that if \u2211i=1naik=\u2211j=1mbjk{displaystyle sum _{i=1}^{n}a_{i}^{k}=sum _{j=1}^{m}b_{j}^{k}}, where ai\u00a0\u2260\u00a0bj are positive integers for all 1\u00a0\u2264\u00a0i\u00a0\u2264\u00a0n and 1\u00a0\u2264\u00a0j\u00a0\u2264\u00a0m, then m+n\u00a0\u2265\u00a0k. The equal sum of like powers formula is often abbreviated as\u00a0(k,\u00a0m,\u00a0n).Small examples with m=n=k2{displaystyle m=n={frac {k}{2}}} (related to generalized taxicab number) include 594+1584=1334+1344{displaystyle 59^{4}+158^{4}=133^{4}+134^{4}} (known to Euler) and 36+196+226=106+156+236{displaystyle 3^{6}+19^{6}+22^{6}=10^{6}+15^{6}+23^{6}} (found by K. Subba Rao in 1934).The conjecture implies in the special case of m = 1 that if\u2211i=1naik=bk{displaystyle sum _{i=1}^{n}a_{i}^{k}=b^{k}}(under the conditions given above) then n\u00a0\u2265\u00a0k\u00a0\u2212\u00a01.For this special case of m\u00a0=\u00a01, some of the known solutions satisfying the proposed constraint with  n\u00a0\u2264\u00a0k, where terms are positive integers, hence giving a partition of a power into like powers, are:[3]k = 333 + 43 + 53 = 63.k = 4958004 + 2175194 + 4145604 = 4224814,  (Roger Frye, 1988)304 + 1204 + 2724 + 3154 = 3534, (R. Norrie, 1911)Fermat’s Last Theorem states that for k = 4 the conjecture is true.k = 5275 + 845 + 1105 + 1335 = 1445,  (Lander, Parkin, 1966)75 + 435 + 575 + 805 + 1005 = 1075, (Sastry, 1934, third smallest)k = 6(None known. As of 2002, there are no solutions whose final term is \u2264 730000.[4] )k = 71277 + 2587 + 2667 + 4137 + 4307 + 4397 + 5257 = 5687,  (M. Dodrill, 1999)k = 8908 + 2238 + 4788 + 5248 + 7488 + 10888 + 11908 + 13248 = 14098,  (Scott Chase, 2000)k \u2265 9(None known.)Current status[edit]It is not known if the conjecture is true, or if nontrivial solutions exist that would be counterexamples, such as ak + bk = ck + dk for k\u00a0\u2265\u00a05.[5][6]Trivial solutions include certain cases with composite exponents k, e.g. k\u00a0=\u00a06, as for such k\u00a0=\u00a0p\u00a0*\u00a0q it is possible to have solutions (ap)q + (bp)q = (aq)p + (bq)p, for positive integers a and b.See also[edit]References[edit]External links[edit]  "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/lander-parkin-and-selfridge-conjecture\/#breadcrumbitem","name":"Lander, Parkin, and Selfridge conjecture"}}]}]